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# What is the greatest value of n such that 30!6n is an integer?

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Math Expert
Joined: 02 Sep 2009
Posts: 56251
What is the greatest value of n such that 30!6n is an integer?  [#permalink]

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29 Apr 2016, 16:07
2
6
00:00

Difficulty:

45% (medium)

Question Stats:

61% (01:08) correct 39% (01:54) wrong based on 148 sessions

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What is the greatest value of n such that 30!/6^n is an integer?

A. 11
B. 12
C. 13
D. 14
E. 15

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Math Expert
Joined: 02 Aug 2009
Posts: 7764
Re: What is the greatest value of n such that 30!6n is an integer?  [#permalink]

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29 Apr 2016, 21:04
5
2
Bunuel wrote:
What is the greatest value of n such that 30!/6^n is an integer?

A. 11
B. 12
C. 13
D. 14
E. 15

This Q basically asks us the number of 6s in 30!..
and to find 6s in 30!, we require to find the biggest prime number of 6 in 30!..

so number of 3s in 30! = $$[\frac{30}{3}]+[\frac{30}{3^2}]+[\frac{30}{3^3}] = 10+3+1 = 14$$
ans D
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Re: What is the greatest value of n such that 30!6n is an integer?  [#permalink]

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25 Jul 2016, 06:08
Bunuel wrote:
What is the greatest value of n such that 30!/6^n is an integer?

A. 11
B. 12
C. 13
D. 14
E. 15

$$6^n$$ can be written as $$(2*3)^n$$ = $$2^n *3^n$$

Now if we can figure how many 2 and 3 are present in 30 ! then we can have them removed. Cancelling(dividing) n number of 2 and 3 will remove n number of 6 (because 6 is a multiple of 2 and 3)
Therefore removing $$2^n$$and$$3^n$$will remove $$6^n$$ and thus we can get the highest number of n that can be removed from 30!

2's in 30! =$$2^{26}$$
3's in 30!= $$3^{14}$$
so we can easily remove $$2^{14} *3^{14}$$ from 30!
so $$6^{14}$$ can be removed
therefore n=14
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Posts: 13
What is the greatest value of n such that 30!6n is an integer?  [#permalink]

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25 Jul 2016, 07:02
chetan2u wrote:
Bunuel wrote:
What is the greatest value of n such that 30!/6^n is an integer?

A. 11
B. 12
C. 13
D. 14
E. 15

This Q basically asks us the number of 6s in 30!..
and to find 6s in 30!, we require to find the biggest prime number of 6 in 30!..

so number of 3s in 30! = $$[\frac{30}{3}]+[\frac{30}{3^2}]+[\frac{30}{3^3}] = 10+3+1 = 14$$
ans D

I knew that we could use $$[\frac{n}{5^1}]+[\frac{n}{5^2}]+[\frac{n}{5^3}] ...$$ for trailing of zeros of n!

But is it true for counting any prime number of n! ?
Math Expert
Joined: 02 Sep 2009
Posts: 56251
Re: What is the greatest value of n such that 30!6n is an integer?  [#permalink]

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25 Jul 2016, 08:51
1
matsuda wrote:
chetan2u wrote:
Bunuel wrote:
What is the greatest value of n such that 30!/6^n is an integer?

A. 11
B. 12
C. 13
D. 14
E. 15

This Q basically asks us the number of 6s in 30!..
and to find 6s in 30!, we require to find the biggest prime number of 6 in 30!..

so number of 3s in 30! = $$[\frac{30}{3}]+[\frac{30}{3^2}]+[\frac{30}{3^3}] = 10+3+1 = 14$$
ans D

I knew that we could use $$[\frac{n}{5^1}]+[\frac{n}{5^2}]+[\frac{n}{5^3}] ...$$ for trailing of zeros of n!

But is it true for counting any prime number of n! ?

Yes, you can find the power of a prime in factorial this way. Check the links below.

Check Trailing Zeros Questions and Power of a number in a factorial questions in our Special Questions Directory.

Hope it helps.
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Re: What is the greatest value of n such that 30!6n is an integer?  [#permalink]

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28 Jul 2016, 11:38
Here is what i did=>
t get the number of 6 we need to count the number of 3's as the number of 2 will always be adequate.
=> 3*1
=> 3*2
=> 3^2*1
=>3*4
=>3*5
=>3^2*2
=>3*7
=>3*8
=>3^3
=>3*10

Number of three's=> 14
Smash that D
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Re: What is the greatest value of n such that 30!6n is an integer?  [#permalink]

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29 Jul 2016, 09:30
30! needs to be divisible by 2 and 3 simultaneously

How many 2's are there ? so that the division leads to be an integer

30/2 + 30/(2^2)+30/(2^3)+30(2^4) = 15+7+3+1=26

How many 3's are there ? so that the division leads to be an integer

30/3 + 30/(3^2)+30/(3^3)= 10+3+1=14

How many 3's and 2's are there so that the division leads to be an integer

Minimum or common between 26 and 14. Answer is 14.

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Re: What is the greatest value of n such that 30!6n is an integer?  [#permalink]

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24 Aug 2018, 06:36
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Re: What is the greatest value of n such that 30!6n is an integer?   [#permalink] 24 Aug 2018, 06:36
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